THEORY OF APPROXIMATION. 101 



expansion may also be considered as being three times as 

 great as the linear expansion. For if the increase of tem- 

 perature expands a bar of metal in the ratio of i to i 4- a, 

 and the expansion be equal in all directions, then a cube 

 of the same metal would expand as i to (i+a) 3 , or as 

 i to i + 3 a + 3 a 2 + a 3 . When a is a very small quantity 

 the third term 3 a 2 will be imperceptible, and still more so 

 the fourth term* a 3 . The coefficients of expansion of 

 solids are in fact so small, and so imperfectly determined, 

 that physicists seldom take into account their second and 

 higher powers. 



It is an universal and important result of these prin- 

 ciples that all very small errors may be assumed to vary 

 in simple proportion to their causes ; a new reason why, in 

 eliminating errors, we should first of all make them as 

 small as possible. Let us suppose, with De Morgan, that 

 there is a right-angled triangle of which the two sides 

 containing the right angle are really of the lengths 3 and 

 4, so that the hypothenuse is v /3 2 + 4 2 or 5. Now if in 

 two measurements of the first side we commit slight 

 errors, making it successively 4*001 and 4*002, then calcu- 

 lation will give the lengths of the hypothenuse as almost 

 exactly 5*0008 and 5*00016, so that the error in the 

 hypothenuse will seem to vary in simple proportion to 

 that of the side, although it does not really do so with 

 perfect exactness a . The logarithm of a number does 

 not vary in proportion to that number nevertheless we 

 should find the difference between the logarithms of the 

 numbers 100000 and 100001 to be almost exactly equal to 

 that between the numbers 100001 and 100002. It is thus 

 a general rule that very small differences between suc- 

 cessive values of a function are approximately proportional 

 to the small differences of the variable quantity. 



a De Morgan's ' Differential Calculus.' 



